The discovery of symmetries in nature has played a fundamental role in the understanding of physical phenomena. In most of the cases, such symmetries provide a zeroth-order approximation and require some kind of breaking in such a way to make the theoretical predictions consistent with the experimental data. In particular, CPT symmetry and active Lorentz covariance, which certainly constitute two of the fundamental pillars of modern physics need to be probed experimentally to the highest precision. Recently, such efforts have been intensified, both from the theoretical and observational points of view. One of the reasons is that most of the proposals to describe the microstructure and the dynamics of space-time (i. e. theories of quantum gravity) suggest the possibility that space is no longer a continuum, but rather has a granular structure, at distances of the order of the Planck length. In this way one could entertain the risk that such symmetries could present minute violations at standard model enegies, suppressed by a scale of the order of the Planck mass. Up to now, violations have not been detected and the bounds upon the parameters that codify such violations are attaining always increasing sensitivities. Even the persistence of such results will have a positive effect: one hand they will constitute a very strong experimental confirmation of such symmetries. On the other hand, those result can provide both experimental guidance and strong constraints to the different contending theories of quantum gravity once the corresponding limits have been performed in order to make contact with the range of energies which are accessible to us. In this talk we present an overview and a summary of some topics related to these investigations, trying to highlight the work developed in Mexico, with an eye of motivating further collaborations.